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W. Hugh Woodin

Publications and source records attributed to W. Hugh Woodin.

11 recordsLinked to original sources
arXiv2026

The Axiom of Real Determinacy and the Axiom of Real Blackwell Determinacy

We show that the Axiom of Real Determinacy $\mathsf{AD}_{\mathbb{R}}$ and the Axiom of Real Blackwell Determinacy $\mathsf{Bl}\text{-}\mathsf{AD}_{\mathbb{R}}$ are equivalent in $\mathsf{ZF}$+$\mathsf{DC}$. This answers the question of Löwe [15, Question 53]. While we do not know whether they are equivalent in $\mathsf{ZF}$+$\mathsf{AC}_ω (\mathbb{R})$, we show that the theories $\mathsf{ZF}$+$\mathsf{AD}_{\mathbb{R}}$ and $\mathsf{ZF}$+$\mathsf{AC}_ω (\mathbb{R})$+$\mathsf{Bl}\text{-}\mathsf{AD}_{\mathbb{R}}$ are equiconsistent.

math.LO↗
arXiv2019

Mice with finitely many Woodin cardinals from optimal determinacy hypotheses

We prove the following result which is due to the third author. Let $n \geq 1$. If $\boldsymbolΠ^1_n$ determinacy and $Π^1_{n+1}$ determinacy both hold true and there is no $\boldsymbolΣ^1_{n+2}$-definable $ω_1$-sequence of pairwise distinct reals, then $M_n^\#$ exists and is $ω_1$-iterable. The proof yields that $\boldsymbolΠ^1_{n+1}$ determinacy implies that $M_n^\#(x)$ exists and is $ω_1$-iterable for all reals $x$. A consequence is the Determinacy Transfer Theorem for arbitrary $n \geq 1$, namely the statement that $\boldsymbolΠ^1_{n+1}$ determinacy implies $\Game^{(n)}(<ω^2 - \boldsymbolΠ^1_1)$ determinacy.

math.LO↗
arXiv2018

Open class determinacy is preserved by forcing

The principle of open class determinacy is preserved by pre-tame class forcing, and after such forcing, every new class well-order is isomorphic to a ground-model class well-order. Similarly, the principle of elementary transfinite recursion $\text{ETR}_Γ$ for a fixed class well-order $Γ$ is preserved by pre-tame class forcing. The full principle ETR itself is preserved by countably strategically closed pre-tame class forcing, and after such forcing, every new class well-order is isomorphic to a ground-model class well-order. Meanwhile, it remains open whether ETR is preserved by all forcing, including the forcing merely to add a Cohen real.

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arXiv2018

The universal finite set

We define a certain finite set in set theory $\{x\midφ(x)\}$ and prove that it exhibits a universal extension property: it can be any desired particular finite set in the right set-theoretic universe and it can become successively any desired larger finite set in top-extensions of that universe. Specifically, ZFC proves the set is finite; the definition $φ$ has complexity $Σ_2$, so that any affirmative instance of it $φ(x)$ is verified in any sufficiently large rank-initial segment of the universe $V_θ$; the set is empty in any transitive model and others; and if $φ$ defines the set $y$ in some countable model $M$ of ZFC and $y\of z$ for some finite set $z$ in $M$, then there is a top-extension of $M$ to a model $N$ in which $φ$ defines the new set $z$. Thus, the set shows that no model of set theory can realize a maximal $Σ_2$ theory with its natural number parameters, although this is possible without parameters. Using the universal finite set, we prove that the validities of top-extensional set-theoretic potentialism, the modal principles valid in the Kripke model of all countable models of set theory, each accessing its top-extensions, are precisely the assertions of S4. Furthermore, if ZFC is consistent, then there are models of ZFC realizing the top-extensional maximality principle.

math.LO↗
arXiv2015

Determinacy and Jónsson cardinals in $L(\mathbb{R})$

Assume $\mathsf{ZF}+\mathsf{AD}+V=L(\mathbb{R})$ and let $κ<Θ$ be an uncountable cardinal. We show that $κ$ is Jónsson, and that if $\mathrm{cof}(κ)=ω$ then $κ$ is Rowbottom. We also establish some other partition properties.

math.LO↗
arXiv2002

Beyond $\underTildeΣ^2_1$ absoluteness

There have been many generalizations of Shoenfield's Theorem on the absoluteness of $Σ^1_2$ sentences between uncountable transitive models of $\mathrm{ZFC}$. One of the strongest versions currently known deals with $Σ^2_1$ absoluteness conditioned on $\mathrm{CH}$. For a variety of reasons, from the study of inner models and from simply combinatorial set theory, the question of whether conditional $Σ^2_2$ absoluteness is possible at all, and if so, what large cardinal assumptions are involved and what sentence(s) might play the role of $\mathrm{CH}$, are fundamental questions. This article investigates the possiblities for $Σ^2_2$ absoluteness by extending the connections between determinacy hypotheses and absoluteness hypotheses.

math.LO↗
arXiv1997

The Jensen covering property

An optimal extension of the Jensen covering lemma, within the limits imposed by Prikry forcing, is proved. If L[E] is an "iterable" weasel with no measurable cardinals, then either L[E] has "indiscernibles", or every uncountable set of ordinals is contained in a set in L[E] of the same cardinality. (The terms "iterable" and "indiscernibles" are made precise in the paper.) Most importantly, there is no hypothesis explicitly limiting the large cardinals which are consistent in L[E].

math.LO↗
arXiv1995

Delta-1-2 Without Sharps

We show that the supremum of the lengths of boldface Delta-1-2 prewellorderings of the reals can be Aleph-2, with Aleph-1 inac- cessible to reals, assuming only the consistency of an inaccessible.

math.LO↗
arXiv1995

The complexity of the reals in inner models of set theory

The usual definition of the set of constructible reals is $Σ^1_2$. This set can have a simpler definition if, for example, it is countable or if every real is constructible. H. Friedman asked if the set of constructible reals can be analytic or even Borel in a nontrivial way. A related problem was posed by K. Prikry: can there exist a nonconstructible perfect set of constructible reals? The main result of this paper is a negative answer to Friedman's question. In fact we prove that if $M$ is an inner model of set theory and the set of reals in $M$ is analytic then either all reals are in $M$ or else $\aleph _1^M$ is countable. We also extend this result to higher levels of the projective hierarchy under appropriate large cardinal assumptions. Concerning Prikry's problem we show that the answer is negative if "perfect" is replaced by "superperfect" but that it can be positive if "constructible" is replaced by "belonging to some inner model $M$".

math.LO↗